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Full text of "Scientific Papers - Vi"

358.
PROBLEMS IN THE CONDUCTION OF HEAT.
[Philosophical Magazine, Vol. xxii. pp. 381—396, 1911.]
THE general equation for the conduction of heat in a uniform medium
may be written
dv _ d'2v    d*v    d2v _ _2                                            ,_.
v representing temperature. The coefficient (v) denoting dirTusibility is omitted for brevity on the right-hand of (1). It can always be restored by consideration of " dimensions."
Kelvin* has shown how to build up a variety of special solutions, applicable to an infinite medium, on the basis of Fourier's solution for a point-source. A few examples are quoted almost in Kelvin's words:
I. Instantaneous simple point-source; a quantity Q of heat suddenly generated at the point (0, 0, 0) at time £ = 0, and left to diffuse through an infinite homogeneous solid:
4. _    ^£_____                                                        /O\
STT"/-^..................................(*>
where r2 = $2 + 2/2 -H £2. [The thermal capacity is supposed to be unity.] Verify that
/•CO
vdoc dy dz — 4nr I    vr'*dr=Qi Jo
and that  v = 0  when t — 0;  unless also x — 0, y = 0, z = 0.   Every other solution is obtainable from this by summation.
II.    Constant simple point-source, rate q:
The formula within the brackets shows how this obvious solution is derivable from (2).
* " Compendium of Fourier Mathematics, &c.," Enc. Brit. 1880; Collected Papers, Vol. n. p. 44.
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